Question

Approximate the number of hours of daylight in Honolulu, Hawaii $21°{18}^{\text{'}}$north latitude, for the following dates:

(a) Summer solstice $\left(i=23.5°\right)$

(b) Vernal equinox $\left(i=0°\right)$

(c) July $4\left(i=22°{48}^{\text{'}}\right)$

Short answer

(a) The number of hours of daylight in summer solstice is, $13$hours and $18$minutes.

(b) The number of hours of daylight in the vernal equinox is $12$ hours.

(c) The number of hours of daylight in July is $13$hours and$15$ minutes.

Step-by-step solution

Step 1. Given data 

We have to find the number of hours of daylight in Honolulu, Hawaii $21°18\text{'}$north latitude in Summer solstice$(i=23.5°)$,Vernal equinox $(i=0°)$and on July $4$$(i=22°48\text{'})$

Step 2. Number of hours of daylight in the summer solstice 

(a) To find the number of hours of daylight we have to use the formula,

$D=24\left[1-\frac{{\cos }^{-1}(\tan i\cdot \tan \theta )}{\pi }\right]......\left(1\right)$

Substitute $i=23.5°$and $\theta =21°18\text{'}$north latitude in this equation,

$\begin{array}{rl}D& =24\left[1-\frac{{\cos }^{-1}(\tan \left({23.5}^{\circ }\right)\tan \left({21}^{\circ }{18}^{\mathrm{\prime }}\right))}{\pi }\right]\\ & =24\left[1-\frac{{\cos }^{-1}\left(0.1695\right)}{\pi }\right]\\ & =24\left[1-\frac{80.2413}{\pi }\right]\end{array}$

Since $80.2413$is in degrees, convert it into radians by multiplying by $\frac{\pi }{180}$.

$80.2413\left(\frac{\pi }{180}\right)=1.40$

Substitute $1.40$radians for $80.2413$degrees in the equation.

localid="1647591160741" $\begin{array}{rl}D& =24\left[1-\frac{1.40}{\pi }\right]\\ & =24[1-0.446]\\ & =24\left[0.554\right]\\ & =13.3\end{array}$

Therefore, the number of hours of daylight in summer solstice is $13.3$hours that is,$13$hours and $18$minutes.

Step 3. Number of hours of daylight in the vernal equinox  

(b) To find the number of hours of daylight in the vernal equinox, substitute $i=0°$and $\theta =21°18\text{'}$in equation (1)

$\begin{array}{rl}D& =24\left[1-\frac{{\cos }^{-1}(\tan \left(0^{\circ }\right)\tan \left({21}^{\circ }{18}^{\mathrm{\prime }}\right))}{\pi }\right]\\ & =24\left[1-\frac{{\cos }^{-1}\left(0\right)}{\pi }\right]\\ & =24\left[1-\frac{90}{\pi }\right]\end{array}$

Since $90$is in degrees, convert it into radians by multiplying by $\frac{\mathrm{\pi }}{180}$.

$90\left(\frac{\pi }{180}\right)=1.57$

Substitute $1.57$radians for $90$degrees in the equation.

$\begin{array}{rl}D& =24[1-\frac{1.57}{\pi }]\\ & =24[1-0.5]\\ & =24\left[0.5\right]\\ & =12\end{array}$

Therefore, the number of hours of daylight in the vernal equinox is $12$hours.

Step 4. Number of hours of daylight on July 4

(c) To find the number of hours of daylight on July $4$, substitute localid="1647592512468" $i=22°48\text{'}$and$\theta =21°18\text{'}$......equation (1)

$$\begin{gathered} D=24\left[1-\frac{{\cos }^{-1}\left(\tan \left({22}^{\circ }{48}^{\mathrm{\prime }}\right)\tan \left({21}^{\circ }{18}^{\mathrm{\prime }}\right)\right)}{\pi }\right] \\ =24\left[1-\frac{{\cos }^{-1}\left(0.1639\right)}{\pi }\right] \\ =24\left[1-\frac{80.57}{\pi }\right] \end{gathered}$$

We have to convert $80.57$into degree, for that divide the minute part by $60$.

$80.57\left(\frac{\pi }{180}\right)=1.4061$

localid="1647593049482" $$\begin{gathered} D=24\left[1-\frac{1.4061}{\pi }\right] \\ =24[1-0.4476] \\ =24\left[0.5524\right] \\ =13.26 \end{gathered}$$

Therefore, the number of hours of daylight in July is $13.26$hours, that is $13$hours and $15$minutes.